The republican elephant represents the enormous fat assed mugwumps of the 19'th century party. Heavyset is understating the elephantine buld of a collosus like Bet a million Gates or Jim Fisk who bought and sold legislators in gross.
The democratic donkey was for the braying rhetorical style of the southern slave state senators. Othern than Spermin' Thurman the best known of these was Senator Foghorn Leghorn who also played himself in a series of humorous Warner Brothers cartoons.